Problem: Given two real numbers $p>1$ and $q>1$ so that $\frac{1}{p} + \frac{1}{q} = 1$ and $pq = 4$, what is $q$?
Explanation: Solving $pq = 4$ for $p$ we see that $p = \frac{4}{q}$. Plugging this into $\frac{1}{p} + \frac{1}{q} = 1$ we then get \[ \frac{q}{4} + \frac{1}{q} = 1 \Rightarrow q^2 - 4q +4 = 0 .\] Factoring this, we then see that  \[ (q-2)(q-2) = 0 \] which means that $q = \boxed{2}$